Mathematical objects of a given type often come in families depending on continuous parameters. These parameters are generally called moduli. So in a sense there are as many moduli spaces as types of mathematical objects, and for this reason moduli spaces form a cross-section of many domains of mathematics.

Already Riemann was able to count the "number of free parameters" defining a Riemann surface of fixed genus. In the formally precise terminology of modern algebraic geometry, he was counting the dimension of the moduli space of Riemann surfaces. Traditionally, one studies these spaces via deformation theory (degenerations and compactifications - this often involves interesting combinatorics) or by uniformization theory (Teichmüller space).

Very surprisingly, these and other more complicated moduli spaces (e.g., of vector bundles, of stable maps, etc.) were discovered in recent years to play an important role in mathematical physics, especially in the theory of quantum strings, which strives to the unification of quantum field theory and the theory of gravity.

Currently, moduli spaces are studied from three perspectives, which we will now describe in more detail.

Initially, moduli spaces were introduced and investigated in order to understand better the objects they parametrize. It turned out that moduli spaces can be used as important tools in proofs of classical results. Often it requires a deep insight to see how a moduli space can be employed to obtain a result that only deals with individual objects. The 1983 proof of the Mordell and Shafarevich conjectures by G. Faltings is a famous example; here the moduli space of abelian varieties plays a decisive role. As it was thus realized that moduli spaces are significant mathematical objects, mathematicians started to study them in their own right. Their intrinsic beauty also contributed to the flourishing of this subject.

Another application of moduli spaces is to consider them as a tool to construct interesting varieties. There are various arithmetic conjectures about varieties that are inaccessible in general. By making use of the fact that a moduli space is parametrizing certain structures it becomes possible to prove some of these conjectures for these special varieties. For instance we can attach an $L$-function to an algebraic variety over $\mathbb Q$ and certain conjectures about its analytic properties can be stated, but we can prove them only in very few cases, and these cases are usually modular varieties. The most spectacular example is the theorem of Wiles that an elliptic curve over $\mathbb Q$ in some sense occurs in a suitable modular curve. These modular curves are the simplest examples of Shimura varieties.

In global Riemannian geometry one considers spaces of isometry classes of complete Riemannian manifolds, defined by specific bounds on the geometry. The questions of interest here include the triviality of such spaces (rigidity) and the structure of their boundaries (e.g., Gromov-Hausdorff convergence), among others. One also studies minimal submanifold immersions or isospectral metrics by looking at their moduli spaces.

The second perspective is to view moduli spaces as a way of generating new geometries. As an example from differential geometry, we mention the Weil-Petersson metric which is a natural Kähler metric on moduli spaces of Calabi-Yau manifolds. In the case of three-dimensional Calabi-Yau manifolds this metric satisfies additional properties which lead to the notion of a special Kähler metric. This geometry was discovered first by physicists in an entirely different context. There it arose as a constraint of extended supersymmetry in four-dimensional supergravity Yang-Mills theories. The link between these two appearances of special Kähler geometry is provided by string theory. The low-energy limit of a ten-dimensional superstring theory compactified down to four dimensions with a Calabi-Yau threefold is identical to the supergravity Yang-Mills theory.

Another example is furnished by the so-called Frobenius manifolds. Physicists discovered that moduli spaces of topological and conformal field theories come up together with a new structure: their tangent vectors can be multiplied as elements of an algebra. After a suitable axiomatization, it was realized that several other constructions lead to the same structure. In particular it emerges on the unfolding spaces of isolated singularities (Kyoji Saito et al.), and the cohomology spaces of certain differential graded algebras (S. Barannikov and M. Kontsevich).

The most surprising and much studied phenomenon, again predicted by physicists, is the famous mirror symmetry. It is expressed in the existence of its morphisms between Frobenius manifolds given by totally different constructions, for example, quantum cohomology (genus zero Gromov-Witten invariants) and extended moduli spaces of Calabi-Yau manifolds.

Mirror symmetry is closely connected with studying degenerations which lie at the boundary of a moduli space. It was recognized recently that moduli spaces may have "invisible" boundary strata which parametrize non-commutative varieties in the sense of Alain Connes. This is a very promising new direction of research.

Finally, the third perspective originates from a fundamental idea about constructing invariants of geometric spaces. The idea is to assign to a geometric space a moduli space and to prove that standard invariants of the moduli space are actually invariants of the geometric space.

As an example of this we mention Donaldson and Seiberg-Witten invariants of 4-manifolds. The gauge-theoretic moduli spaces arising in these theories are spaces of solutions of certain partial differential equations defined in terms of geometric objects such as connections and spinors, modulo a large group of gauge symmetries. These moduli spaces provide invariants of smooth 4-manifolds, and by dimensional reduction and gluing formulae, invariants of 3-manifolds in the form of Floer homology theories. An intensive study of gauge theoretic moduli spaces was initiated in the early eighties, with Donaldson's famous result on obstructions to the existence of smooth structures on certain classes of 4-manifolds. More recent results have uncovered deep connections between moduli spaces of Donaldson and Seiberg-Witten invariants of 4-manifolds, and between Seiberg-Witten and Gromov-Witten invariants of symplectic 4-manifolds.

In the study of all kinds of moduli spaces from all of these perspectives, explicit formulas are often given by identities involving modular functions (i.e., functions on moduli spaces). The appearance of modular forms is ubiquitous in the theory of $L$-functions, but very surprisingly, they are also involved in many identities arising from mirror symmetry, vertex algebras and statistical mechanics.